Game apparatuses and methods for use in teaching the addition, subtraction, multiplication, and division of positive and negative numbers

ABSTRACT

Game apparatuses and methods for assisting in teaching the addition, subtraction, multiplication, and division of positive and, especially, negative numbers, are based on the Null Theory of Adding, Subtracting, Multiplying, and Dividing Positive and Negative Numbers. An exemplary game apparatus comprises (a) a plurality of positive units, (b) a plurality of negative units, (c) a demarcated playing zone, and (d) a means for measuring the number of free positive units and free negative units within the demarcated playing zone. Within the demarcated playing zone, free positive units combine with free negative units zone to form null units, with each null unit comprising an equal number of positive and negative units. Each null unit preferably comprises one positive unit and one negative unit.

CROSS REFERENCE TO RELATED APPLICATION

[0001] This application is a continuation-in-part of U.S. patentapplication Ser. No. 10/264,875, filed Oct. 4, 2002, entitled GameApparatuses and Methods for Use in Teaching the Addition and Subtractionof Positive and Negative Numbers, which application is incorporatedherein in its entirety by reference.

FIELD OF THE INVENTION

[0002] The present invention relates to educational game apparatuses andmethods for use in teaching the addition, subtraction, multiplication,and division of positive and, especially, negative numbers. The gameapparatuses and methods of the present invention enable students to seeand understand a theory for adding, subtracting, multiplying, anddividing positive and negative numbers.

[0003] Description of the Prior Art

[0004] A description of the prior art is set forth in U.S. Pat. No.1,294,126, U.S. Pat. No. 3,094,792, U.S. Pat. No. 3,229,388, U.S. Pat.No. 3,410,002, U.S. Pat. No. 3,414,986, U.S. Pat. No. 3,452,454, U.S.Pat. No. 3,935,649, U.S. Pat. No. 4,177,681, U.S. Pat. No. 5,474,455,U.S. Pat. No. 6,089,871, and U.S. Pat. No. 6,413,099, which patents areincorporated herein in their entireties by reference.

[0005] As evidenced by the above-cited patents, educational gameapparatuses and methods exist for teaching mathematical concepts.However, teaching the addition, subtraction, multiplication, anddivision of positive and, especially, negative numbers usually entailsstudents learning by rote the rules of adding, subtracting, multiplying,and dividing positive and negative numbers without ever understandingthe rhyme or reason behind what they are doing. Other students,unfortunately, never learn the rules and, for them, mathematics becomesa dreaded black hole.

SUMMARY OF THE INVENTION

[0006] Accordingly, a technique is needed for teaching the addition,subtraction, multiplication, and division of positive and, especially,negative numbers that clearly explains a cogent theory behind the rules.

[0007] The apparatuses and methods of the present invention for teachingthe addition, subtraction, multiplication, and division of positive andnegative numbers solve the above need. More specifically, the presentinvention is based on the Null Theory of Adding, Subtracting,Multiplying, and Dividing Positive and Negative Numbers (hereinafterreferred to as the “Null Theory”). According to the Null Theory, thenatural state of a given environment is the null state. In the nullstate, the environment is in perfect balance and appears to be devoid ofany matter. However, the environment is, in fact, composed of aplurality of null units, with each null unit being, in turn, composed ofa positive unit and a negative unit. The environment can be disturbed byintroducing (i.e., adding) into it one or more positive units or one ormore negative units. The environment can also be disturbed by removing(i.e., subtracting) from it one or more positive units or one or morenegative units. In the latter case, if there are not enough freepositive units available to be removed from the environment, asufficient number of null units are split to obtain the desired numberof positive units to be removed from the environment. When a null unitis split and the positive unit thereof is removed from the environment,a negative unit is left behind in the environment. (Hence, the foregoingexplanation of the Null Theory clarifies and visually demonstrates thereason behind the rule that the subtraction of a positive number +X isequal to the addition of a negative number (i.e., −(+X)=+(−X)).)Likewise, if there are not enough free negative units available to beremoved from the environment, a sufficient number of null units aresplit to obtain the desired number of negative units to be removed fromthe environment. When a null unit is split and the negative unit thereofis removed from the environment, a positive unit is left behind in theenvironment. (Accordingly, the foregoing explanation of the Null Theoryclarifies and visually demonstrates the reason behind the rule that thesubtraction of a negative number −X is equal to the addition of apositive number (i.e., −(−X)=+(+X)).)

[0008] Another aspect of the Null Theory is that only excess positiveunits or excess negative units remain in the free state within theenvironment. For example, if there are 5 free positive units in theenvironment and if 3 negative units are introduced into the environment(as is the case in the mathematical expression 5+(−3)), the 3 negativeunits will combine with 3 of the free positive units to form 3 nullunits, leaving only 2 free positive units in the environment.

[0009] With the Null Theory in mind, in one embodiment of the presentinvention, the game apparatus employed to teach the addition,subtraction, multiplication, and division of positive and negativenumbers comprises (a) a plurality of positive units, (b) a plurality ofnegative units, and (c) a demarcated playing environment or zone. Thepositive units and the negative units are adapted to reversibly attachto or be associated with one another to form null units, with each nullunit comprising at least one positive unit and at least one negativeunit and the number of positive units and the number of negative unitsper null unit being equal. (As used in the specification and claims, theterms “attached to” and “associated with” both mean that the objects inquestion either can be physically reversibly held together or can bepositioned in a manner such that the objects appear to have an affinityfor or relationship with one another.) Preferably, each null unitcomprises just one positive unit and just one negative unit.

[0010] The demarcated playing zone is typically an integral part of aplaying surface.

[0011] Generally, the game apparatus further comprising a first meansfor measuring the number of units selected from the group consistingpositive units, negative units, and combinations thereof, with the firstmeasuring means desirably being located on the playing surface and,preferably, within the demarcated playing zone. The purpose of the firstmeasuring means is to measure the degree that the demarcated playingzone has been disturbed from the null state. The first measuring means,which can be a scale for weighting the positive and/or negative units,is ideally an axis marked in substantially equal units from 0 to M andin substantially equal units from 0 to N, where M is a positive wholeinteger, N is a negative whole integer, and substantially each of thepositive units is adapted to reversibly attach to or be associated witha unit from 0 to M on the axis on the playing surface, and substantiallyeach of the negative units is adapted to reversibly attach to or beassociated with a unit from 0 to N on the axis on the playing surface.While M can be virtually any positive integer, M is typically a wholepositive integer from 5 to 50, more typically from 10 to 25, and mosttypically from 10 to 20. Similarly, while N can be virtually anynegative integer, N is commonly a whole negative number from −5 to −50,more commonly from −10 to −25, and most commonly from −10 to −20.Usually, M equals the absolute value of N.

[0012] In another preferred embodiment of the invention, the apparatusfurther comprises a second means for measuring the number of unitsselected from the group consisting positive units, negative units, andcombinations thereof, with the second measuring means desirably beinglocated on the playing surface and, preferably, outside the demarcatedplaying zone. The purpose of the second measuring means is to act as acheck point or zone to ensure that (a) the correct number of positiveunits and/or negative units are being transported into the demarcatedplaying zone and (b) the correct number of positive units and/ornegative units have been removed from the demarcated playing zone. Thesecond measuring means, which can also be a scale for weighting thepositive units and/or the negative units, is ideally an axis marked insubstantially equal units from 0 to P, where P is a positive wholeinteger and substantially each of the positive and negative units isadapted to reversibly attach to or be associated with a unit from 0 to Pon the axis on the playing surface. While P can be virtually anypositive integer, P is typically a whole positive integer from 10 to100, more typically from 20 to 50, and most typically from 20 to 40.Usually, P equals M plus the absolute value of N.

[0013] It is also preferred that the game apparatus further comprise aplurality of means for reversibly holding a plurality of the null unitslocated within the demarcated playing zone.

[0014] The game apparatus preferably also comprises an additional meansfor reversibly holding at least one positive unit and an additionalmeans for reversibly holding at least one negative unit. Morepreferably, the game apparatus further comprises an additional means forreversibly holding a plurality of positive units and an additional meansfor reversibly holding a plurality of negative units. In one embodimentof the invention, the additional holding means comprises a groovedivided into two sections, with one section adopted to hold a pluralityof the positive units and the other section adopted to hold a pluralityof the negative units. Most preferably, the additional positive unitholding means and the additional negative unit holding means are anintegral part of the playing surface and are located outside thedemarcated playing zone.

[0015] The playing surface of the game apparatus can be a screen of anelectrical unit (such as a computer screen, a television screen, etc), asurface of a game board, a surface of a blackboard or other writingsurface, a Velcro® surface, a static electricity charged surface, amagnetized surface, a magnetizable surface, or the surface of any otherapparatus or device capable of displaying the demarcated playing zone,the null units, the positive units, the negative units, and the meansfor showing the number of free positive units and/or free negative unitswithin the demarcated playing zone.

[0016] The game apparatus of the present invention is employed inconjunction with methods that utilize the principles of the Null Theory.In particular, in the case of adding and subtracting positive andnegative numbers, the method of the present invention comprises thefollowing steps:

[0017] Step A:

[0018] Play begins in a null state where a demarcated playing zonecomprises a plurality of null units, with each null unit comprising atleast one positive unit reversibly attached to at least one negativeunit, the number of positive and negative units per null unit beingequal.

[0019] Step B:

[0020] Take the first mathematical operation in a mathematical problemX₁ S_(m) X_(m) S_(n) X_(n) . . . S_(z) X_(z) where the mathematicalexpressions X₁, X_(m), X_(n), and X^(z) are independently selected fromthe group consisting of positive numbers, negative numbers, andcombinations thereof, the mathematical operators S_(m), S_(n), and S_(z)are independently selected from the group consisting of the additionoperation and the subtraction operation, m is selected from the groupconsisting of 0 and 2, n is selected from the group consisting of 0 and3, provided that if m is 0, n is 0, and z is selected from the groupconsisting of 0 and whole integers greater that 3, provided that if m is0, z is 0, and perform the first mathematical operation X₁ as follows:

[0021] 1. If X₁ is a positive number, then the mathematical operation ofstep (B) comprises moving or adding X₁ free positive units to thedemarcated playing zone.

[0022] 2. If X₁ is a negative number, then the mathematical operation ofstep (B) comprises moving or adding the absolute value of X₁ freenegative units to the demarcated playing zone.

[0023] Alternatively, the mathematical operation indicated by the firstmathematical expression X₁ can be performed as follows:

[0024] 1. If X₁ is a positive number, then the mathematical operation ofstep (B) can be accomplished by breaking apart or separating X₁ nullunits that are within the demarcated playing zone into X₁ free positiveunits and into X₁ free negative units and removing or subtracting the X₁free negative units from the demarcated playing zone.

[0025] 2. If X₁ is a negative number, then the mathematical operation ofstep (B) can be accomplished by breaking apart or separating theabsolute value of X₁ null units that are within the demarcated playingzone into the absolute value of X₁ free positive units and into theabsolute value of X₁ free negative units and removing or subtracting theabsolute value of X₁ free positive units from the demarcated playingzone.

[0026] Step C:

[0027] Take the second mathematical operation S₂X₂ in the mathematicalproblem, where S₂ is selected from the group consisting of the additionoperation and the subtraction operation and X₂ is selected from thegroup consisting of positive and negative numbers, and perform themathematical operation indicated by S₂X₂ as follows:

[0028] 1. When S₂ is an addition operation and X₂ is a positive number(e.g., +(+3)), then step (C) comprises moving or adding X₂ free positiveunits to the demarcated playing zone and, if there are any free negativeunits within the demarcated playing zone, then step (C) furthercomprises combining up to X₂ free negative units that are already insidethe demarcated playing zone with up to the X₂ free positive units thatwere moved into the demarcated playing zone.

[0029] 2a. When S₂ is a subtraction operation and X₂ is a positivenumber (e.g., −(+3)), then step (C) comprises removing or subtracting X₂free positive units from the demarcated playing zone and, if there arenot X₂ free positive units within the demarcated play zone to removefrom the demarcated zone, then step (C) further comprises breaking apartor separating enough null units that are within the demarcated playingzone to obtain up to the required X₂ free positive units and removingthe X₂ free positive units from the demarcated playing zone.

[0030] 2b. Alternatively, when S₂ is a subtraction operation and X₂ is apositive number (e.g., −(+3)), then step (C) can also be accomplished bymoving or adding X₂ free negative units to the demarcated playing zoneand, if there are any free positive units within the demarcated playingzone, then step (C) further comprises combining up to X₂ free positiveunits that are already inside the demarcated playing zone with up to theX₂ free negative units that were moved into the demarcated playing zone.

[0031] 3a. When S₂ is an addition operation and X₂ is a negative number(e.g., +(−3)), then step (C) comprises moving or adding the absolutevalue of X₂ free negative units to the demarcated playing zone and, ifthere are any free positive units within the demarcated playing zone,then step (C) further comprises combining up to the absolute value of X₂free positive units that are already within the demarcated playing zonewith up to the absolute value of X₂ free negative units that were movedinto the demarcated playing zone.

[0032] 3b. Alternatively, when S₂ is an addition operation and X₂ is anegative number (e.g., +(−3)), then step (C) can also be accomplished byremoving or subtracting X₂ free positive units from the demarcatedplaying zone and, if there are not X₂ free positive units within thedemarcated play zone to remove from the demarcated zone, then step (C)further comprises breaking apart or separating enough null units thatare within the demarcated playing zone to obtain up to the required X₂free positive units and removing the X₂ free positive units from thedemarcated playing zone.

[0033] 4. When S₂ is a subtraction operation and X₂ is a negative number(e.g., −(−3)), then step (C) comprises removing or subtracting theabsolute value of X₂ free negative units from the demarcated playingzone and, if there are not enough absolute value of X₂ free negativeunits within the demarcated play zone to remove from the demarcatedplaying zone, then step (C) further comprises breaking apart orseparating enough null units that are within the demarcated playing zoneto obtain up to the required absolute value of X₂ free negative unitsand removing the absolute value of X₂ free negative units from thedemarcated playing zone.

[0034] Step D:

[0035] Repeat step (C) for each of the remaining mathematical operationsS_(n)X_(n) through S_(z)X_(z) in the problem, where X_(n) through X_(z)are independently selected from the group consisting of positivenumbers, negative numbers, and combinations thereof, and themathematical operators S_(n), through S_(z) are independently selectedfrom the group consisting of the addition operation and the subtractionoperation.

[0036] As the above discussion of adding and subtracting positive andnegative numbers demonstrates, when the operation sign and the sign ofthe mathematical expression are different, the same result followswhether the operation sign or the sign of the mathematical expression istreated as the mathematical operator as long as the other sign istreated as the sign of the unit. To illustrate, for the mathematicalexpression −(+3), the same result is obtained when the “−” sign istreated as the mathematical operator and the “+” sign is treated asdesignating the type of unit (i.e., 3 positive units) as when the “+”sign is treated as the mathematical operator and the “−” sign is treatedas designating the type of unit (i.e., 3 negative units). This principalof interchangeability of signs, which is also applicable to multiplyingand dividing positive and negative numbers, follows from the wellestablished rule that ab=ba or, in our specific case, (+)(−)=(−)(+).Accordingly, students learning how to add, subtract, multiply, anddivide positive and negative numbers do not have to memorize which signis to be treated as the operator sign and which sign has to be treatedas the unit sign. The students just have to remember that if they treatone of the signs as the operator sign, they must treat the other sign asthe unit sign.

[0037] In the case of multiplying positive and negative numbers, themethod of the present invention comprises the following steps:

[0038] Step A:

[0039] Play begins in a null state where a demarcated playing zonecomprises a plurality of null units, with each null unit comprising atleast one positive unit reversibly attached to at least one negativeunit, the number of positive and negative units per null unit beingequal.

[0040] Step B

[0041] In this scenario, X₁ is a multiplication expression(S_(x)M₁)(SyN₁), where S_(x) and S_(y) are independently selected fromthe group consisting of a positive sign and a negative sign, M₁ is theabsolute value of a number, and N₁ is the absolute value of a number.The multiplication expression (S_(x)M₁)(S_(y)N₁) is performed asfollows:

[0042] 1. If S_(x) and S_(y) are positive signs, move or add theabsolute value of (M₁)(N₁) positive units to the demarcated playingzone. Since the demarcated playing zone is initially in the null state,there are initially no free positive units or free negative units in thedemarcated playing zone. Accordingly, after adding the absolute value of(M₁)(N₁) positive units to the demarcated playing zone as required bystep (B)(1), the total number of free positive units in the demarcatedplaying zone is the absolute value of (M₁)(N₁). 2a. If S_(x) is apositive sign and S_(y) is a negative sign, remove or subtract theabsolute value of (M₁)(N₁) positive units from the demarcated playingzone. Since the demarcated playing zone is initially in the null state,the free positive units are removed from the demarcated playing zone bytaking the absolute value of (M₁)(N₁) null units that are within thedemarcated playing zone, breaking or separating them into theirconstituent positive units and negative units, and removing the absolutevalue of (M₁)(N₁) positive units from the demarcated playing zone. Thus,the absolute value of (M₁)(N₁) negative units are left behind in thedemarcated playing zone.

[0043] 2b. Alternatively, when S_(x) is a positive sign and S_(y) is anegative sign, then step (B) can also be accomplished by moving oradding the absolute value of (M₁)(N₁) negative units to the demarcatedplaying zone. Since the demarcated playing zone is initially in the nullstate, there are initially no free positive units or free negative unitsin the demarcated playing zone. Accordingly, after adding the absolutevalue of (M₁)(N₁) negative units to the demarcated playing zone asrequired by step (B)(2b), the total number of free negative units in thedemarcated playing zone is the absolute value of (M₁)(N₁).

[0044] 3a. If S_(x) is a negative sign and S_(y) is a positive sign,move or add the absolute value of (M₁)(N₁) negative units to thedemarcated playing zone. Since the demarcated playing zone is initiallyin the null state, there are initially no free positive units or freenegative units in the demarcated playing zone. Accordingly, after addingthe absolute value of (M₁)(N₁) negative units to the demarcated playingzone as required by step (B)(3a), the total number of free negativeunits in the demarcated playing zone is the absolute value of (M₁)(N₁).

[0045] 3b. Alternatively, when S_(x) is a negative sign and S_(y) is apositive sign, then step (B) can also be accomplished by removing orsubtracting the absolute value of (M₁)(N₁) positive units from thedemarcated playing zone. Since the demarcated playing zone is initiallyin the null state, the free positive units are removed from thedemarcated playing zone by taking the absolute value of (M₁)(N₁) nullunits that are within the demarcated playing zone, breaking orseparating them into their constituent positive units and negativeunits, and removing the absolute value of (M₁)(N₁) positive units fromthe demarcated playing zone. Thus, the absolute value of (M₁)(N₁)negative units are left behind in the demarcated playing zone.

[0046] 4. If S_(x) and S_(y) are negative signs, remove or subtract theabsolute value of (M₁)(N₁) negative units from the demarcated playingzone. Since the demarcated playing zone is initially in the null state,the free negative units are removed from the demarcated playing zone bytaking the absolute value of (M₁)(N₁) null units that are within thedemarcated playing zone, breaking or separating them into theirconstituent positive units and negative units, and removing the absolutevalue of (M₁)(N₁) negative units from the demarcated playing zone. Thus,the absolute value of (M₁)(N₁) positive units are left behind in thedemarcated playing zone.

[0047] In the case of dividing positive and negative numbers, the methodof the present invention comprises the following steps:

[0048] Step A:

[0049] Play begins in a null state where a demarcated playing zonecomprises a plurality of null units, with each null unit comprising atleast one positive unit reversibly attached to at least one negativeunit, the number of positive and negative units per null unit beingequal.

[0050] Step B

[0051] In this scenario, X₁ is a division expression(S_(xx)M₁)/(S_(yy)N₁), where S_(xx) and S_(yy) are independentlyselected from the group consisting of a positive sign and a negativesign, M₁ is the absolute value of a number, and N₁ is the absolute valueof a number. The division expression (S_(xx)M₁)/(S_(yy)N₁) is performedas follows:

[0052] 1. If S_(xx) and S_(yy) are positive signs, move or add theabsolute value of (M₁)/(N₁) positive units to the demarcated playingzone. Since the demarcated playing zone is initially in the null state,there are initially no free positive units or free negative units in thedemarcated playing zone. Accordingly, after adding the absolute value of(M₁)/(N₁) positive units to the demarcated playing zone as required bystep (B)(1), the total number of free positive units in the demarcatedplaying zone is the absolute value of (M₁)/(N₁).

[0053] 2a. If S_(xx) is a positive sign and S_(yy) is a negative sign,remove or subtract the absolute value of (M₁)/(N₁) positive units fromthe demarcated playing zone. Since the demarcated playing zone isinitially in the null state, the free positive units are removed fromthe demarcated playing zone by taking the absolute value of (M₁)/(N₁)null units that are within the demarcated playing zone, breaking orseparating them into their constituent positive units and negativeunits, and removing the absolute value of (M₁)/(N₁) positive units fromthe demarcated playing zone. Thus, the absolute value of (M₁)/(N₁)negative units are left behind in the demarcated playing zone.

[0054] 2b. Alternatively, when S_(xx) is a positive sign and S_(yy) is anegative sign, then step (B) can also be accomplished by moving oradding the absolute value of (M₁)/(N₁) negative units to the demarcatedplaying zone. Since the demarcated playing zone is initially in the nullstate, there are initially no free positive units or free negative unitsin the demarcated playing zone. Accordingly, after adding the absolutevalue of (M₁)/(N₁) negative units to the demarcated playing zone asrequired by step (B)(2b), the total number of free negative units in thedemarcated playing zone is the absolute value of (M₁)/(N₁).

[0055] 3a. If S_(xx) is a negative sign and S_(yy) is a positive sign,move or add the absolute value of (M₁)/(N₁) negative units to thedemarcated playing zone. Since the demarcated playing zone is initiallyin the null state, there are initially no free positive units or freenegative units in the demarcated playing zone. Accordingly, after addingthe absolute value of (M₁)/(N₁) negative units to the demarcated playingzone as required by step (B)(3), the total number of free negative unitsin the demarcated playing zone is the absolute value of (M₁)/(N₁).

[0056] 3b. Alternatively, when S_(xx) is a negative sign and S_(yy) is apositive sign, then step (B) can also be accomplished by removing orsubtracting the absolute value of (M₁)/(N₁) positive units from thedemarcated playing zone. Since the demarcated playing zone is initiallyin the null state, the free positive units are removed from thedemarcated playing zone by taking the absolute value of (M₁)/(N₁) nullunits that are within the demarcated playing zone, breaking orseparating them into their constituent positive units and negativeunits, and removing the absolute value of (M₁)/(N₁) positive units fromthe demarcated playing zone. Thus, the absolute value of (M₁)/(N₁)negative units are left behind in the demarcated playing zone.

[0057] 4. If S_(xx) and S_(yy) are negative signs, remove or subtractthe absolute value of (M₁)/(N₁) negative units from the demarcatedplaying zone. Since the demarcated playing zone is initially in the nullstate, the free negative units are removed from the demarcated playingzone by taking the absolute value of (M₁)/(N₁) null units that arewithin the demarcated playing zone, breaking or separating them intotheir constituent positive units and negative units, and removing theabsolute value of (M₁)/(N₁) negative units from the demarcated playingzone. Thus, the absolute value of (M₁)/(N₁) positive units are leftbehind in the demarcated playing zone.

[0058] When the mathematical equation includes the addition and/orsubtraction of one or more multiplication and/or division expressions bythemselves and/or together with the addition and/or subtraction of oneor more positive and/or negative numbers, each operation can be solvedin the order that it appears in the equation. Alternatively, each of themultiplication and division expressions in the mathematical equation canbe solved first, thereby converting the mathematical equation into onecontaining only addition and/or subtraction operations, and solving theresulting mathematical equation as discussed above.

[0059] As noted with respect to the game apparatus, in the methods ofthe present invention, each null unit preferably comprises one positiveunit reversibly attached to or associated with one negative unit.

[0060] It is also preferred that the methods further comprise the stepof measuring the number of the free positive units and the number of thefree negative units within the demarcated playing zone. In one preferredversion of the methods of the present invention, the measurement isperformed by the step of placing the free positive units that are withinthe demarcated playing zone along the positive portion of an axis markedwith substantially equal spaces from 0 to M and the step of placing thefree negative units that are within the demarcated playing zone alongthe negative portion of an axis marked with substantially equal spacesfrom 0 to N, where M and N are as defined above. While the positiveportion of the axis preferably forms a continuum with the negativeportion of the axis, the present invention includes the embodiment wherethere is a separate positive axis having substantially equal spaces from0 to M and a separate negative axis having substantially equal spacesfrom 0 to N, where M and N are as previously defined.

[0061] Furthermore, it is preferred that the methods of the presentinvention also comprise the step of measuring the number of freepositive units and the number of free negative units that are to beadded to or that have been removed from the demarcated playing zone. Inone desirable version of the methods of the present invention, themeasurement is performed by the step of placing the free positive unitsor the free negative units that are to be added to or that have beentaken out of the demarcated playing zone along a second axis that islocated outside the demarcated playing zone. The second axis is markedwith substantially equal spaces from 0 to P, where P is as previouslydefined.

[0062] While the foregoing apparatuses and methods can be used to teachyoung children who are just learning to add and subtract only positivenumbers, when teaching the addition and subtraction of only positivenumbers, it is sufficient to use a simpler game apparatus such as onecomprising (a) at least one means for measuring unit increments, (b) aplurality of means for indicating a single unit, and (c) a means forholding the plurality of single unit indicating means in slideablerelationship to the measuring means. For instance, the plurality ofsingle unit indicating means can comprise a plurality of beads, theholding means can comprise a dowel, with the beads being slideablymounted on the dowel, and the game apparatus can comprises two measuringmeans (such as two rulers), with each measuring means being position sothat the game apparatus can be played with equal facility by both rightand left handed players.

[0063] For a fuller understanding of the nature and advantages of themathematical game apparatuses and methods of the present invention,reference should be made to the ensuing detailed description taken inconjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Exemplary game apparatuses of the present invention are shown inthe drawings where:

[0065]FIG. 1 is a top view of a playing surface employed in the methodsof the present invention,

[0066]FIG. 2 is an isometric view of a null unit employed in the methodsof the present invention showing a negative sign on one of the twoconstituent components of the null unit;

[0067]FIG. 3 is another isometric view of a null unit employed in themethods of the present invention showing a positive sign on one of thetwo constituent components of the null unit;

[0068]FIG. 4 is an isomeric view of a null unit separated into its twoconstituent halves, namely, the positive component or unit and thenegative component or unit;

[0069]FIG. 5 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the firstmathematical operation of the mathematical problem of Example 1;

[0070]FIG. 6 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the first throughsecond mathematical operations of the mathematical problem of Example 1;

[0071]FIG. 7 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the first throughthird mathematical operations of the mathematical problem of Example 1;

[0072]FIG. 8 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the first throughfourth mathematical operations of the mathematical problem of Example 1;

[0073]FIG. 9 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the first throughfifth mathematical operations of the mathematical problem of Example 1;

[0074]FIG. 10 is a perspective view of the front side of an apparatuswithin the scope of the present invention for teaching the addition andsubtraction of just positive whole numbers;

[0075]FIG. 11 is a perspective view of the reverse side of the apparatusshown in FIG. 10 for teaching the addition and subtraction of justpositive whole numbers;

[0076]FIG. 12 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the multiplicationproblem of Example 2;

[0077]FIG. 13 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the multiplicationproblem of Example 3;

[0078]FIG. 14 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the multiplicationproblem of Example 4;

[0079]FIG. 15 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the multiplicationproblem of Example 5;

[0080]FIG. 16 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the divisionproblem of Example 6;

[0081]FIG. 17 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the divisionproblem of Example 7;

[0082]FIG. 18 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the divisionproblem of Example 8;

[0083]FIG. 19 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the divisionproblem of Example 9;

[0084]FIG. 20 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the divisionoperation of the mathematical problem of Example 10;

[0085]FIG. 21 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the multiplicationoperation of the mathematical problem of Example 10;

[0086]FIG. 22 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the firstmathematical operation of the mathematical problem of Example 10;

[0087]FIG. 23 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the first throughsecond mathematical operations of the mathematical problem of Example10;

[0088]FIG. 24 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the first throughthird mathematical operations of the mathematical problem of Example 10;

[0089]FIG. 25 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the first throughfourth mathematical operations of the mathematical problem of Example10;

[0090]FIG. 26 is a top view of a playing surface employed in the methodsof the present invention modified in accordance with the first throughfifth mathematical operations of the mathematical problem of Example 10;and

[0091]FIG. 27 is a top view of another embodiment of the playing surfaceemployed in the methods of the present invention.

[0092] It should be noted that the same numbers in the figures representthe same element of the game apparatuses of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

[0093]FIG. 1 shows an embodiment of the present invention in which thegame apparatus 1 comprises a playing surface 2, a demarcated playingenvironment or zone 4 located within the playing surface 2, a pluralityof free positive units 6 (located outside the demarcated playing zone 4)removably attached to the playing surface 2, a plurality of freenegative units 8 (located outside the demarcated playing zone 4)removably attached to the playing surface 2, a plurality of null units10 (located inside the demarcated playing zone 4) removably attached tothe playing surface 2, and an axis 12 (located within the demarcatedplaying zone 4) numbered from −10 to 10.

[0094] The playing surface 2 can be a game board, a computer screen, atelevision screen, a liquid crystal display screen, or any other meansfor displaying the demarcated playing zone 4, the free positive units 6,the free negative units 8, the null units 10, and the axis 12.

[0095] The area of the demarcated playing zone 4 can comprise a portionof the area of the playing surface 2 as shown in FIG. 1 or can comprisethe entire area of the playing surface 2 (not shown). When the area ofdemarcated playing zone 4 constitutes the entire area of the playingsurface 2, the free positive units 6 and the free negative units 8either can be held in essentially any container (such as a cup, a bowl,a bag, a pocket, a groove, etc.) or can be loose.

[0096] In the version of the invention shown in FIG. 1, the freepositive units 6 and the free negative units 8 are removably attached tothe playing surface 2 by snuggly fitting into depressions 14 in theplaying surface 2. Other means for removably attaching the positiveunits 6 and the negative units 8 to the playing surface 2 include, butare not limited to, Velcro, snap fittings (such as used on clothing oron Lego® building blocks), screw fittings, magnetism, gravity, etc. Ofcourse, when the playing surface 2 is a computer screen or otheressentially two-dimensional means for displaying the playing surface 2,the positive units 6 and negative units 8 are merely displayed on ormerely displayed as being associated with the essentiallytwo-dimensional means for displaying the playing surface 2.

[0097] Similarly, in the version of the invention shown in FIG. 1, thenull units 10 are also removably attached to the demarcated playing zone4 by snuggly fitting into depressions 16 in the demarcated playing zone4. Likewise, other means for removably attaching the null units 10 tothe demarcated playing zone 4 also include, but are not limited to,Velcro, snap fittings (such as used on clothing or on Lego® buildingblocks), screw fittings, magnetism, gravity, etc. In addition, when theplaying surface 2 is a computer screen or other essentiallytwo-dimensional means for displaying the playing surface 2, the nullunits 10 are merely displayed on or merely displayed as being associatedwith the essentially two-dimensional means for displaying the playingsurface 2.

[0098] In the embodiment of the invention shown in FIG. 1, the amount offree positive units 6 and free negative units 8 in the demarcatedplaying zone 4 are measured along the axis 12. As shown in FIG. 1, theaxis 12 is numbered from 10 to −10. However, the axis 12 could just aswell be numbered from M to N, where M is a positive integer and N is anegative integer. Preferably, M is a positive integer from 5 to 50, morepreferably from 10 to 25, and most preferably from 10 to 20. Similarly,N preferably is a negative integer from −5 to −50, more preferably from−10 to −25, and most preferably from −10 to −20. Usually, M is equal tothe absolute value of N. Furthermore, other means such as a weightingscale, a unit counter, etc. can be used to measure the number of freepositive units 6 and free negative units 8 within the demarcated playingzone 4. Also, the measuring means can be located inside or outside thedemarcated playing zone 4 and, in fact, on or off the playing surface 2.When the playing surface 2 is the display media used in conjunction witha computerized version of one or more of the methods of the presentinvention, all that need be shown on the display media is the number offree positive units 6 and/or the number of free negative units 8 thatare within the demarcated playing zone 4.

[0099] As shown in more detail in FIGS. 2 and 3, each null unit 10consists of a positive unit 6 removably attached to a negative unit 8.However, the null unit 10 can consist of a plurality of positive units 6and a plurality of negative units 8 removably attached together,provided that the number of positive units 6 and the number of negativeunits 8 per null unit 10 are equal. Nevertheless, the null unit 10preferably consists of a single positive unit 6 removably attached to asingle negative unit 8.

[0100] In the embodiment of the invention shown in FIG. 4, the positiveunit 6 has a neck 18 that is adapted to snuggly fit into an opening 20in the negative unit 8 to form the null unit 10. Other means forremovably attaching the positive units 6 to the negative units 8 to formthe null units 10 include, but are not limited to, Velcro, snap fittings(such as used on clothing or on Lego® building blocks), screw fittings,magnetism, etc. In addition, when the playing surface 2 is a computerscreen or other essentially two-dimensional means for displaying theplaying surface 2, the positive unit 6 and negative unit 8 portions ofeach null unit 10 are merely displayed in close proximity (i.e., asbeing associated with one another) on the essentially two-dimensionalmeans for displaying the playing surface 2.

[0101] The game apparatus of the present invention is employed inconjunction with methods for adding, subtracting, multiplying, anddividing positive and negative numbers. In particular, the method of thepresent invention for adding and subtracting positive and negativenumbers comprises the following steps:

[0102] Step A:

[0103] Play begins in a null state where a demarcated playing zone 4comprises a plurality of null units 10, with each null unit 10comprising at least one positive unit 6 reversibly attached to orassociated with at least one negative unit 8, the number of positiveunits 6 and negative units 8 per null unit 10 being equal.

[0104] Step B:

[0105] Take the first mathematical operation in a mathematical problemX₁ S_(m) X_(m) S_(n) X_(n) . . . S_(z) X_(zw)here the mathematicalexpressions X₁, X_(m), X_(n), and X_(z) are independently selected fromthe group consisting of positive numbers, negative numbers, andcombinations thereof, the mathematical operators S_(m), S_(n), and S_(z)are independently selected from the group consisting of the additionoperation and the subtraction operation, m is selected from the groupconsisting of 0 and 2, n is selected from the group consisting of 0 and3, provided that if m is 0, n is 0, and z is selected from the groupconsisting of 0 and whole integers greater that 3, provided that if m is0, z is 0, and perform the first mathematical operation X₁ as follows:

[0106] 1. If X₁ is a positive number, then the mathematical operation ofstep (B) comprises moving or adding X₁ free positive units 6 to thedemarcated playing zone 4.

[0107] 2. If X₁ is a negative number, then the mathematical operation ofstep (B) comprises moving or adding the absolute value of X₁ freenegative units 8 to the demarcated playing zone 4.

[0108] Alternatively, the mathematical operation indicated by the firstmathematical expression X₁ can be performed as follows:

[0109] 1. If X₁ is a positive number, then the mathematical operation ofstep (B) can be accomplished by breaking apart or separating X₁ nullunits 10 that are within the demarcated playing zone 4 into X₁ freepositive units 6 and into X₁ free negative units 8 and removing orsubtracting the X₁ free negative units 8 from the demarcated playingzone 4.

[0110] 2. If X₁ is a negative number, then the mathematical operation ofstep (B) can be accomplished by breaking apart or separating theabsolute value of X₁ null units 10 that are within the demarcatedplaying zone 4 into the absolute value of X₁ free positive units 6 andinto the absolute value of X₁ free negative units 8 and removing orsubtracting the absolute value of X₁ free positive units 6 from thedemarcated playing zone.

[0111] Step C:

[0112] Take the second mathematical operation S₂X₂ in the mathematicalproblem, where S₂ is selected from the group consisting of the additionoperation and the subtraction operation and X₂ is selected from thegroup consisting of positive and negative numbers, and perform themathematical operation indicated by S₂X₂ as follows:

[0113] 1. When S₂ is an addition operation and X₂ is a positive number(e.g., +(+3)), then step (C) comprises moving or adding X₂ free positiveunits 6 to the demarcated playing zone 4 and, if there are any freenegative units 8 within the demarcated playing zone 4, then step (C)further comprises combining up to X₂ free negative units 8 that arealready inside the demarcated playing zone 4 with up to the X₂ freepositive units 6 that were moved into the demarcated playing zone 4.

[0114] 2a. When S₂ is a subtraction operation and X₂ is a positivenumber (e.g., −(+3)), then step (C) comprises removing or subtracting X₂free positive units 6 from the demarcated playing zone 4 and, if thereare not X₂ free positive units 6 within the demarcated play zone 4 toremove from the demarcated zone 4, then step (C) further comprisesbreaking apart or separating enough null units 10 that are within thedemarcated playing zone 4 to obtain up to the required X₂ free positiveunits 6 and removing the X₂ free positive units 6 from the demarcatedplaying zone 4.

[0115] 2b. Alternatively, when S₂ is a subtraction operation and X₂ is apositive number (e.g., −(+3)), then step (C) can also be accomplished bymoving or adding X₂ free negative units 8 to the demarcated playing zone4 and, if there are any free positive units 6 within the demarcatedplaying zone 4, then step (C) further comprises combining up to X₂ freepositive units 6 that are already inside the demarcated playing zone 4with up to the X₂ free negative units 8 that were moved into thedemarcated playing zone.

[0116] 3a. When S₂ is an addition operation and X₂ is a negative number(e.g., +(−3)), then step (C) comprises moving or adding the absolutevalue of X₂ free negative units 8 to the demarcated playing zone 4 and,if there are any free positive units 6 within the demarcated playingzone 4, then step (C) further comprises combining up to the absolutevalue of X₂ free positive units 6 that are already within the demarcatedplaying zone 4 with up to the absolute value of X₂ free negative units 8that were moved into the demarcated playing zone 4.

[0117] 3b. Alternatively, when S₂ is an addition operation and X₂ is anegative number (e.g., +(−3)), then step (C) can also be accomplished byremoving or subtracting X₂ free positive units 6 from the demarcatedplaying zone 4 and, if there are not X₂ free positive units 6 within thedemarcated play zone 4 to remove from the demarcated zone 4, then step(C) further comprises breaking apart or separating enough null units 10that are within the demarcated playing zone 4 to obtain up to therequired X₂ free positive units 6 and removing the X₂ free positiveunits 6 from the demarcated playing zone 4.

[0118] 4. When S₂ is a subtraction operation and X₂ is a negative number(e.g., −(−3)), then step (C) comprises removing or subtracting theabsolute value of X₂ free negative units 8 from the demarcated playingzone 4 and, if there are not enough absolute value of X₂ free negativeunits 8 within the demarcated play zone 4 to remove from the demarcatedplaying zone 4, then step (C) further comprises breaking apart orseparating enough null units 10 that are within the demarcated playingzone 4 to obtain up to the required absolute value of X₂ free negativeunits 8 and removing the absolute value of X₂ free negative units 8 fromthe demarcated playing zone 4.

[0119] Step D:

[0120] Repeat step (C) for each of the remaining mathematical operationsS_(n)X_(n) through S_(z)X_(z) in the problem, where X_(n) through X_(z)are independently selected from the group consisting of positivenumbers, negative numbers, and combinations thereof, and themathematical operators S_(n) through S_(z) are independently selectedfrom the group consisting of the addition operation and the subtractionoperation.

[0121] The method of the present invention for adding and subtractingpositive and negative numbers is further illustrated in the followingExample 1, which is intended to demonstrate, and not limit, the presentinvention:

EXAMPLE 1 Solve the Equation for x: 10+(−20)−(−15)−5+3=x

[0122] Initially, the demarcated playing zone 4 is in the null state asshown in FIG. 1. In the null state, there are no free positive units 6or free negative units 8 in the demarcated playing zone 4.

[0123] The first step in solving the above equation is to write theequation in full as follows:

+10+(−20)−(−15)−(+5)+(+3)=x

[0124] In discussing the solution to foregoing equation, each element ofthe equation shall be identified as follows:

X ₁ S ₂ X ₂ S ₃ X ₃ S ₄ X ₄ S ₅ X ₅ +10+(−20)−(−15)−(+5)+(+3)=x

[0125] With respect to the mathematical expression X₁, X₁ is the initialdisturbed state of the demarcated playing zone 4. As shown in FIG. 5,the initial disturbed state of the demarcated playing zone 4 is achievedby moving or adding 10 free positive units 6 a to the demarcated playingzone 4. The 10 free positive units 6 a moved into the demarcated playingzone 4 are measured by placing them in the first 10 positive spacesalong the axis 12.

[0126] Regarding the mathematical operator S₂, S₂ is “+”, i.e., apositive sign, which denotes that the mathematical expression X₂ is tobe transported into (i.e., added to) the demarcated playing zone 4. X₂is −20 (i.e., 20 negative units 8). As shown in FIG. 6, 10 of the 20negative units 8 that are moved or added to the demarcated playing zone4 combine with the 10 free positive units 6 a (see FIG. 5) already inthe demarcated playing zone 4 to form 10 null units 10 b. The remaining10 free negative units 8 b moved into the demarcated playing zone 4 aremeasured by placing them in the first 10 negative spaces along the axis12.

[0127] As to the mathematical operator S₃, S₃ is “−”, i.e., a negativesign, which denotes that the mathematical expression X₃ is to betransported out of (i.e., subtracted from) the demarcated playing zone4. X₃ is −15 (i.e., 15 negative units 8). As shown in FIG. 7, 10 of the15 negative units 8 c that are to be removed from the demarcated playingzone 4 are present in the demarcated playing zone 4 as free negativeunits 8 b of FIG. 6. The remaining 5 negative units 8 cc to betransported out of the demarcated playing zone 4 are obtained by taking5 null units 10 from within the demarcated playing zone 4, breaking orseparating these null units 10 into their component parts, namely, 5free positive units 6 cc and 5 free negative units 8 cc, removing thethus obtained 5 free negative units 8 cc from the demarcated playingzone 4, and placing the remaining 5 free positive units 6 cc in thefirst 5 positive spaces along the axis 12.

[0128] With respect to the mathematical operator S₄, S₄ is “−”, i.e., anegative sign, which denotes that the mathematical expression X₄ is tobe transported out of (i.e., subtracted from) the demarcated playingzone 4. X₄ is +5 (i.e., 5 positive units 6). As shown in FIG. 8, all ofthe 5 positive units 6 d that were moved out of or subtracted from thedemarcated playing zone 4 were already present in the demarcated playingzone 4 as free positive units 6 cc shown in FIG. 7.

[0129] Regarding the mathematical operator S₅, S₅ is “+”, i.e., apositive sign, which denotes that the mathematical expression X₅ is tobe transported into (i.e., added to) the demarcated playing zone 4. X₅is +3 (i.e., 3 positive units 6). As shown in FIG. 9, since there are nofree positive units 6 or free negative units 8 in the demarcated playingzone 4, the 3 free positive units 6 e moved into the demarcated playingzone 4 are measured by placing them in the first 3 positive spaces alongthe axis 12.

[0130] Accordingly, x equals 3.

[0131] It should be noted that the above solution would also be achievedif the above-discussed alternative method of treating the signs wereemployed in solving the equation.

[0132] In the case of multiplying positive and negative numbers, themethod of the present invention comprises the following steps:

[0133] Step A:

[0134] Play begins in a null state where a demarcated playing zone 4comprises a plurality of null units 10, with each null unit 10comprising at least one positive unit 6 reversibly attached to at leastone negative unit 8, the number of positive units 6 and negative units 8per null unit 10 being equal.

[0135] Step B

[0136] In this instance, X₁ is a multiplication expression(S_(x)M₁)(S_(y)N₁), where Sx and S_(y) are independently selected fromthe group consisting of a positive sign and a negative sign, M₁ is theabsolute value of a number, and N₁ is the absolute value of a number.The multiplication expression (S_(x)M₁)(S_(y)N₁) is solved as follows:

[0137] 1. If S_(x) and S_(y) are positive signs, move or add theabsolute value of (M₁)(N₁) positive units 6 to the demarcated playingzone 4. Since the demarcated playing zone 4 is initially in the nullstate, there are initially no free positive units 6 or free negativeunits 8 in the demarcated playing zone 4. Accordingly, after adding theabsolute value of (M₁)(N₁) positive units 6 to the demarcated playingzone 4 as required by step (B)(1), the total number of free positiveunits 6 in the demarcated playing zone 4 is the absolute value of(M₁)(N₁).

[0138] 2a. If S_(x) is a positive sign and S_(y) is a negative sign,remove or subtract the absolute value of (M₁)(N₁) positive units 6 fromthe demarcated playing zone 4. Since the demarcated playing zone 4 isinitially in the null state, the free positive units 6 are removed fromthe demarcated playing zone 4 by taking the absolute value of (M₁)(N₁)null units 10 that are within the demarcated playing zone 4, breaking orseparating them into their constituent positive units 6 and negativeunits 8, and removing the absolute value of (M₁)(N₁) positive units 6from the demarcated playing zone 4. Thus, the absolute value of (M₁)(N₁)negative units 8 are left behind in the demarcated playing zone 4.

[0139] 2b. Alternatively, when S_(x) is a positive sign and S_(y) is anegative sign, the step (B) can also be accomplished by moving or addingthe absolute value of (M₁)(N₁) negative units 8 to the demarcatedplaying zone 4. Since the demarcated playing zone 4 is initially in thenull state, there are initially no free positive units 6 or freenegative units 8 in the demarcated playing zone 4. Accordingly, afteradding the absolute value of (M₁)(N₁) negative units 8 to the demarcatedplaying zone 4 as required by step (B)(2 b), the total number of freenegative units 8 in the demarcated playing zone 4 is the absolute valueof (M₁)(N₁).

[0140] 3a. If S_(x) is a negative sign and S_(y) is a positive sign,move or add the absolute value of (M₁)(N₁) negative units 8 to thedemarcated playing zone 4. Since the demarcated playing zone 4 isinitially in the null state, there are initially no free positive units6 or free negative units 8 in the demarcated playing zone 4.Accordingly, after adding the absolute value of (M₁)(N₁) negative units8 to the demarcated playing zone 4 as required by step (B)(3), the totalnumber of free negative units 8 in the demarcated playing zone 4 is theabsolute value of (M₁)(N₁).

[0141] 3b. Alternatively, when S_(x) is a negative sign and S_(y) is apositive sign, then step (B) can also be accomplished by removing orsubtracting the absolute value of (M₁)(N₁) positive units 6 from thedemarcated playing zone 4. Since the demarcated playing zone isinitially in the null state, the free positive units 6 are removed fromthe demarcated playing zone 4 by taking the absolute value of (M₁)(N₁)null units 10 that are within the demarcated playing zone 4, breaking orseparating them into their constituent positive units 6 and negativeunits 8, and removing the absolute value of (M₁)(N₁) positive units 6from the demarcated playing zone 4. Thus, the absolute value of (M₁)(N₁)negative units 8 are left behind in the demarcated playing zone 4.

[0142] 4. If S_(x) and S_(y) are negative signs, remove or subtract theabsolute value of (M₁)(N₁) negative units 8 from the demarcated playingzone 4. Since the demarcated playing zone 4 is initially in the nullstate, the free negative units 8 are removed from the demarcated playingzone 4 by taking the absolute value of (M₁)(N₁) null units 10 that arewithin the demarcated playing zone 4, breaking or separating them intotheir constituent positive units 6 and negative units 8, and removingthe absolute value of (M₁)(N₁) negative units 8 from the demarcatedplaying zone 4. Thus, the absolute value of (M₁)(N₁) positive units 6are left behind in the demarcated playing zone 4.

[0143] The method of the present invention for multiplying positive andnegative numbers is illustrated in the following Examples 2 through 5,which are also intended to demonstrate, and not limit, the presentinvention:

EXAMPLE 2 Solve the Equation for x: (2)(5)=x

[0144] Initially, the demarcated playing zone 4 is in the null state asshown in FIG. 1. In the null state, there are no free positive units 6or free negative units 8 in the demarcated playing zone 4.

[0145] The first step in solving the above equation is to write theequation in full as follows:

(+2)(+5)=x

[0146] As shown in FIG. 12, the mathematical expression (+2)(+5) issolved by moving or adding (2)(5) or 10 positive units 6 f into thedemarcated playing zone 4. Accordingly, as shown in FIG. 12, x equals+10.

EXAMPLE 3 Solve the Equation for x: (3)(−3)=x

[0147] Initially, the demarcated playing zone 4 is in the null state asshown in FIG. 1. In the null state, there are no free positive units 6or free negative units 8 in the demarcated playing zone 4.

[0148] The first step in solving the above equation is to write theequation in full as follows:

(+3)(−3)=x

[0149] As shown in FIG. 13, the mathematical expression (+3)(−3) issolved by removing or subtracting (3)(3) or 9 positive units 6 g fromthe demarcated playing zone 4. Since there are initially no freepositive units 6 in the demarcated playing zone 4, this is accomplishedby taking 9 null units 10 that are within the demarcated playing zone 4,breaking or separating them into their constituent positive units 6 gand negative units 8 g, and removing 9 positive units 6 g from thedemarcated playing zone 4. Thus, 9 negative units 8 g are left behind inthe demarcated playing zone 4 and, as shown in FIG. 13, x equals −9.

EXAMPLE 4 Solve the Equation for x: (−2)(4)=x

[0150] Initially, the demarcated playing zone 4 is in the null state asshown in FIG. 1. In the null state, there are no free positive units 6or free negative units 8 in the demarcated playing zone 4.

[0151] The first step in solving the above equation is to write theequation in full as follows:

(−2)(+4)=x

[0152] As shown in FIG. 14, the mathematical expression (−2)(+4) issolved by moving or adding (2)(4) or 8 negative units 8 h into thedemarcated playing zone 4. Accordingly, as shown in FIG. 14, x equals−8.

EXAMPLE 5 Solve the Equation for x: (−5)(−2)=x

[0153] Initially, the demarcated playing zone 4 is in the null state asshown in FIG. 1. In the null state, there are no free positive units 6or free negative units 8 in the demarcated playing zone 4.

[0154] It should be noted that the above equation is already written outin full.

[0155] As shown in FIG. 15, the mathematical expression (−5)(−2) issolved by removing or subtracting (5)(2) or 10 negative units 8 i fromthe demarcated playing zone 4. Since there are initially no freenegative units 8 in the demarcated playing zone 4, this is accomplishedby taking 10 null units 10 that are within the demarcated playing zone4, breaking or separating them into their constituent positive units 6 iand negative units 8 i, and removing 10 negative units 8 i from thedemarcated playing zone 4. Thus, 10 positive units 6 i are left behindin the demarcated playing zone 4 and, as shown in FIG. 15, x equals +10.

[0156] It should be noted that, if the above-discussed alternativemethod of treating the signs were employed in solving the multiplicationexpressions of above Examples 2 through 5, the answers would be thesame.

[0157] As for the case of dividing positive and negative numbers, themethod of the present invention comprises the following steps:

[0158] Step A:

[0159] Play begins in a null state where a demarcated playing zone 4comprises a plurality of null units 10, with each null unit 10comprising at least one positive unit 6 reversibly attached to at leastone negative unit 8, the number of positive units 6 and negative units 8per null unit 10 being equal.

[0160] Step B

[0161] In this instance, X₁ is a division expression(S_(xx)M₁)/(S_(yy)N₁), where S_(xx) and S_(yy) are independentlyselected from the group consisting of a positive sign and a negativesign, M₁ is the absolute value of a number, and N₁ is the absolute valueof a number. The division expression (S_(xx)M₁)(S_(yy)N₁) is solved asfollows:

[0162] 1. If S_(xx) and S_(yy) are positive signs, move or add theabsolute value of (M₁)/(N₁) positive units 6 to the demarcated playingzone 4. Since the demarcated playing zone 4 is initially in the nullstate, there are initially no free positive units 6 or free negativeunits 8 in the demarcated playing zone 4. Accordingly, after adding theabsolute value of (M₁)/(N₁) positive units 6 to the demarcated playingzone 4 as required by step (B)(1), the total number of free positiveunits 6 in the demarcated playing zone 4 is the absolute value of(M₁)/(N₁).

[0163] 2a. If S_(xx) is a positive sign and S_(yy) is a negative sign,remove or subtract the absolute value of (M₁)/(N₁) positive units 6 fromthe demarcated playing zone 4. Since the demarcated playing zone 4 isinitially in the null state, the free positive units 6 are removed fromthe demarcated playing zone 4 by taking the absolute value of (M₁)/(N₁)null units 10 that are within the demarcated playing zone 4, breaking orseparating them into their constituent positive units 6 and negativeunits 8, and removing the absolute value of (M₁)/(N₁) positive units 6from the demarcated playing zone 4. Thus, the absolute value of(M₁)/(N₁) negative units 8 are left behind in the demarcated playingzone 4.

[0164] 2b. Alternatively, when S_(xx) is a positive sign and S_(yy) is anegative sign, then step (B) can also be accomplished by moving oradding the absolute value of (M₁)/(N₁) negative units 8 to thedemarcated playing zone 4. Since the demarcated playing zone 4 isinitially in the null state, there are initially no free positive units6 or free negative units 8 in the demarcated playing zone 4.Accordingly, after adding the absolute value of (M₁)/(N₁) negative units8 to the demarcated playing zone 4 as required by step (B)(2b), thetotal number of free negative units 8 in the demarcated playing zone isthe absolute value of (M₁)/(N₁).

[0165] 3a. If S_(xx) is a negative sign and S_(yy) is a positive sign,move or add the absolute value of (M₁)/(N₁) negative units 8 to thedemarcated playing zone 4. Since the demarcated playing zone 4 isinitially in the null state, there are initially no free positive units6 or free negative units 8 in the demarcated playing zone 4.Accordingly, after adding the absolute value of (M₁)/(N₁) negative units8 to the demarcated playing zone 4 as required by step (B)(3), the totalnumber of free negative units 8 in the demarcated playing zone 4 is theabsolute value of (M₁)/(N₁).

[0166] 3b. Alternatively, when S_(xx) is a negative sign and S_(yy) is apositive sign, then step (B) can also be accomplished by removing orsubtracting the absolute value of (M₁)/(N₁) positive units 6 from thedemarcated playing zone 4. Since the demarcated playing zone isinitially in the null state, the free positive units 6 are removed fromthe demarcated playing zone 4 by taking the absolute value of (M₁)/(N₁)null units 10 that are within the demarcated playing zone 4, breaking orseparating them into their constituent positive units 6 and negativeunits 8, and removing the absolute value of (M₁)/(N₁) positive units 6from the demarcated playing zone 4. Thus, the absolute value of(M₁)/(N₁) negative units 8 are left behind in the demarcated playingzone 4.

[0167] 4. If S_(xx) and S_(yy) are negative signs, remove or subtractthe absolute value of (M₁)/(N₁) negative units 8 from the demarcatedplaying zone 4. Since the demarcated playing zone 4 is initially in thenull state, the free negative units 8 are removed from the demarcatedplaying zone 4 by taking the absolute value of (M₁)/(N₁) null units 10that are within the demarcated playing zone 4, breaking or separatingthem into their constituent positive units 6 and negative units 8, andremoving the absolute value of (M₁)/(N₁) negative units 8 from thedemarcated playing zone 4. Thus, the absolute value of (M₁)/(N₁)positive units 6 are left behind in the demarcated playing zone 4.

[0168] The method of the present invention for dividing positive andnegative numbers is illustrated in the following Examples 6 through 9,which are likewise intended to demonstrate, and not limit, the presentinvention:

EXAMPLE 6 Solve the Equation for x: (20)/(5)=x

[0169] Initially, the demarcated playing zone 4 is in the null state asshown in FIG. 1. In the null state, there are no free positive units 6or free negative units 8 in the demarcated playing zone 4.

[0170] The first step in solving the above equation is to write theequation in full as follows:

(+20)/(+5)=x

[0171] As shown in FIG. 16, the mathematical expression (+20)/(+5) issolved by moving or adding (20)/(5) or 4 positive units 6 j to thedemarcated playing zone 4. Accordingly, as shown in FIG. 16, x equals+4.

EXAMPLE 7 Solve the Equation for x: (30)/(−3)=x

[0172] Initially, the demarcated playing zone 4 is in the null state asshown in FIG. 1. In the null state, there are no free positive units 6or free negative units 8 in the demarcated playing zone 4.

[0173] The first step in solving the above equation is to write theequation in full as follows:

(+30)/(−3)=x

[0174] As shown in FIG. 17, the mathematical expression (+30)/(−3) issolved by removing or subtracting (30)/(3) or 10 positive units 6 k fromthe demarcated playing zone 4. Since there are initially no freepositive units 6 in the demarcated playing zone 4, this is accomplishedby taking (30)/(3) or 10 null units 10 that are in the demarcatedplaying zone 4, breaking or separating them into their constituentpositive units 6 k and negative units 8 k, and removing 10 positiveunits 6 k from the demarcated playing zone 4. Thus, 10 negative units 8k are left behind in the demarcated playing zone 4 and, as shown in FIG.17, x equals −10.

EXAMPLE 8 Solve the Equation for x: (−20)/(+4)=x

[0175] Initially, the demarcated playing zone 4 is in the null state asshown in FIG. 1. In the null state, there are no free positive units 6or free negative units 8 in the demarcated playing zone 4.

[0176] The first step in solving the above equation is to write theequation in full as follows:

(−20)/(+4)=x

[0177] As shown in FIG. 18, the mathematical expression (−20)/(+4) issolved by moving or adding (20)/(4) or 5 negative units 8 m to thedemarcated playing zone 4. Accordingly, as shown in FIG. 14, x equals−5.

EXAMPLE 9 Solve the Equation for x: (−200)/(−25)=x

[0178] Initially, the demarcated playing zone 4 is in the null state asshown in FIG. 1. In the null state, there are no free positive units 6or free negative units 8 in the demarcated playing zone 4.

[0179] It should be noted that the above equation is already written outin full.

[0180] As shown in FIG. 19, the mathematical expression (−200)/(−25) issolved by removing or subtracting (200)/(25) or 8 negative units 8 nfrom the demarcated playing zone 4. Since there are initially no freenegative units 8 in the demarcated playing zone 4, this is accomplishedby taking 8 null units 10 that are within the demarcated playing zone 4,breaking or separating them into their constituent positive units 6 nand negative units 8 n, and removing 8 negative units 8 n from thedemarcated playing zone 4. Thus, 8 positive units 6 n are left behind inthe demarcated playing zone 4 and, as shown in FIG. 19, x equals +8.

[0181] It should be noted that, if the above-discussed alternativemethod of treating the signs were employed in solving the divisionexpressions of above Examples 6 through 9, the answers would be thesame.

[0182] As demonstrated in the following Examples 10 and 11, which againare intended to demonstrate, and not limit, the present invention, theabove described methods for adding, subtracting, multiplying, anddividing can be employed to solve a mathematical equation that includesthe addition and/or subtraction of one or more multiplication and/ordivision expressions by themselves and/or together with one or moreaddition and/or subtraction expressions.

EXAMPLE 10 Solve the Equation for x: 9+((−32)/(4))−((−4)(−2))−3+7=x

[0183] Initially, the demarcated playing zone 4 is in the null state asshown in FIG. 1. In the null state, there are no free positive units 6or free negative units 8 in the demarcated playing zone 4.

[0184] The first step in solving the above equation is to write theequation in full as follows:

+9+((−32)/(+4))−((−4)(−2))−(+3)+(+7)=x

[0185] Next, it is preferred to first solve each of the multiplicationand division expressions in the mathematical equation, therebyconverting the mathematical equation into one containing only additionand/or subtraction expressions, and solving the resulting mathematicalexpressions as discussed above.

[0186] Solve the Divisional Expression (−32)/(+4)

[0187] Initially, the demarcated playing zone 4 is in the null state asshown in FIG. 1. In the null state, there are no free positive units 6or free negative units 8 in the demarcated playing zone 4.

[0188] As shown in FIG. 20, the division expression (−32)/(+4) is solvedby moving or adding (32)/(4) or 8 negative units 8 p to the demarcatedplaying zone 4. Accordingly, as shown in FIG. 20, the divisionalexpression (−32)/(+4) equals −8.

[0189] Solve the Multiplication Expression (−4)(−2)

[0190] Initially, the demarcated playing zone 4 is in the null state asshown in FIG. 1. In the null state, there are no free positive units 6or free negative units 8 in the demarcated playing zone 4.

[0191] As shown in FIG. 21, the mathematical expression (−4)(−2) issolved by removing or subtracting (4)(2) or 8 negative units 8 q fromthe demarcated playing zone 4. Since there are initially no freepositive units 6 or free negative units 8 in the demarcated playing zone4, this is accomplished by taking 8 null units 10 that are in thedemarcated playing zone 4, breaking or separating them into theirconstituent positive units 6 q and negative units 8 q, and removing 8negative units 8 q from the demarcated playing zone 4. Thus, 8 positiveunits 6 q are left behind in the demarcated playing zone 4 and, as shownin FIG. 21, the multiplication expression (−4)(−2) equals +8.

[0192] Substitute Solutions to Division and Multiplication Expressionsinto Equation

[0193] Next, the solutions to the division and multiplicationexpressions are substituted into the initial equation to yield thefollowing revised equation:

+9+(−8)−(+8)−(+3)+(+7)=x

[0194] Accordingly, the revised equation only deals with the additionand subtraction of positive and negative numbers and can be solved in amanner analogous to the methodology employed in above Example 1 asfollows:

[0195] Solve the Equation

[0196] In discussing the solution to foregoing equation, each element ofthe equation shall be identified as follows:

X₁ S ₂ X ₂ S ₃ X ₃ S ₄ X ₄ S ₅ X ₅ +9+(−8)−(+8)−(+3)+(+7)=x

[0197] With respect to the mathematical expression X₁, X₁ is the initialdisturbed state of the demarcated playing zone 4. As shown in FIG. 22,the initial disturbed state of the demarcated playing zone 4 is achievedby moving or adding 9 free positive units 6 r to the demarcated playingzone 4. The 9 free positive units 6 r moved into the demarcated playingzone 4 are measured by placing them in the first 9 positive spaces alongthe axis 12.

[0198] Regarding the mathematical operator S₂, S₂ is “+”, i.e., apositive sign, which denotes that the mathematical expression X₂ is tobe transported into (i.e., added to) the demarcated playing zone 4. X₂is −8 (i.e., 8 negative units 8). As shown in FIG. 23, all 8 of the 8negative units 8 that are moved or added to the demarcated playing zone4 combine with the 8 of the 9 free positive units 6 r (see FIG. 22)already in the demarcated playing zone 4 to form 8 null units 10 s.Thus, as shown in FIG. 23, there is only 1 positive unit 6 s left in thedemarcated playing zone 4.

[0199] As to the mathematical operator S₃, S₃ is “−”, i.e., a negativesign, which denotes that the mathematical expression X₃ is to betransported out of (i.e., subtracted from) the demarcated playing zone4. X₃ is +8 (i.e., 8 positive units 6). As shown in FIG. 23, only 1 ofthe 8 positive units 6 that are to be removed or subtracted from thedemarcated playing zone 4 is present in the demarcated playing zone 4 asfree positive unit 6 s. The remaining 7 positive units 6 t to betransported out of the demarcated playing zone 4 are obtained by taking7 null units 10 from within the demarcated playing zone 4, breaking orseparating these 7 null units 10 into their component parts, namely, 7free positive units 6 t and 7 free negative units 8 t, removing the thusobtained 7 free positive units 6 t from the demarcated playing zone 4,and placing the remaining 7 free negative units 8 t in the first 7negative spaces along the axis 12 as shown in FIG. 24.

[0200] With respect to the mathematical operator S₄, S₄ is “−”, i.e., anegative sign, which denotes that the mathematical expression X₄ is tobe transported out of (i.e., subtracted from) the demarcated playingzone 4. X₄ is +3 (i.e., 3 positive units 6). As shown in FIG. 24, noneof the 3 positive units 6 to be removed from the demarcated playing zone4 are present in the demarcated playing zone 4. Accordingly, 3 nullunits 10 that are within the demarcated playing zone 4 are taken, brokenor separated into their 3 constituent positive units 6 u and their 3constituent negative units 8 u, and the 3 positive units 6 u are removedfrom the demarcated playing zone 4 as shown in FIG. 25. The 3 remainingfree negative units 8 u are added to the negative portion of the axis12.

[0201] Regarding the mathematical operator S₅, S₅ is “+”, i.e., apositive sign, which denotes that the mathematical expression X₅ is tobe transported into (i.e., added to) the demarcated playing zone 4. X₅is +7 (i.e., 7 positive units 6). As shown in FIG. 25, there are already10 free negative units 8 in the demarcated playing zone 4. Accordingly,as shown in FIG. 26, the 7 free positive units 6 moved or added to thedemarcated playing zone 4 combine with 7 of the free negative units 8that are already in the demarcated playing zone 4 to form 7 null units10 v. Thus, only 3 negative units 8 v are left in the demarcated playingzone 4.

[0202] Accordingly, x equals −3.

EXAMPLE 11 Solve the Equation for x: 9+((−32)/(4))−((−4)(−2))−3+7=x

[0203] The equation to be solved in Example 11 is identical to theequation solved in Example 10. However, in the present Example 11, eachmathematical expression of the equation is solved sequentially, i.e.,without first solving the division and multiplication expressionsseparately.

[0204] Initially, the demarcated playing zone 4 is in the null state asshown in FIG. 1. In the null state, there are no free positive units 6or free negative units 8 in the demarcated playing zone 4.

[0205] The first step in solving the above equation is to write theequation in full as follows:

+9+((−32)/(+4))−((−4)(−2))−(+3)+(+7)=x

[0206] Solve the Equation

[0207] In discussing the solution to foregoing equation, each element ofthe equation shall be identified as follows:

X₁ S ₂ X ₂ S ₃ X ₃ S ₄ X ₄ S ₅ X ₅+9+((−32)/(+4))−((−4)(−2))−(+3)+(+7)=x

[0208] With respect to the mathematical expression X₁, X₁ is the initialdisturbed state of the demarcated playing zone 4. As shown in FIG. 22,the initial disturbed state of the demarcated playing zone 4 is achievedby moving or adding 9 free positive units 6 r to the demarcated playingzone 4. The 9 free positive units 6 r moved into the demarcated playingzone 4 are measured by placing them in the first 9 positive spaces alongthe axis 12.

[0209] Regarding the mathematical operator S₂, S₂ is “+”, i.e., apositive sign, which denotes that the mathematical expression X₂ is tobe transported into (i.e., added to) the demarcated playing zone 4. X₂is a division expression (−32)/(+4). The division expression X₂ must besolved first and then the resulting addition expression A+B can besolved. The division expression (−32)/(+4) is solved by moving or adding(32)/(4) or 8 negative units 8 p to the demarcated playing zone 4 which,for purposes of this operation, is initially assumed to be in the nullstate. Accordingly, as shown in FIG. 20, the divisional expression(−32)/(+4) equals −8 (i.e., 8 negative units 8 e). Because S₁ is apositive sign, the 8 negative units 8 are moved or added to thedemarcated playing zone 4. As shown in FIG. 23, all 8 of the negativeunits moved into the demarcated playing zone 4 combine with the 8 of the9 free positive units 6 r (see FIG. 22) already in the demarcatedplaying zone 4 to form 8 null units 10 s. Thus, as shown in FIG. 23,there is only 1 positive unit 6 s left in the demarcated playing zone 4.

[0210] As to the mathematical operator S₃, S₃ is “−”, i.e., a negativesign, which denotes that the mathematical expression X₃ is to betransported out of (i.e., subtracted from) the demarcated playing zone4. X₃ is a multiplication expression (−4)(−2). The multiplicationexpression X₃ must be solved first and then the resulting subtractionexpression (A+B)−C can be solved. The multiplication expression (−4)(−2)is solved by removing or subtracting (4)(2) or 8 negative units 8 q fromthe demarcated playing zone 4 which, for purposes of this operation, isinitially assumed to be in the null state. Since there are initially nofree negative units 8 in the demarcated playing zone 4, this isaccomplished by taking 8 null units 10 that are in the demarcatedplaying zone 4, breaking or separating them into their constituentpositive units 6 q and negative units 8 q, and removing 8 negative units8 q from the demarcated playing zone 4. Thus, as shown in FIG. 21, 8positive units 6 q are left behind in the demarcated playing zone 4 and,hence, the multiplication expression (−4)(−2) equals +8. Because S₂ is anegative sign, 8 positive units 8 must be removed or subtracted from thedemarcated playing zone 4. However, as shown in FIG. 23, only 1 of the 8positive units 6 that are to be removed or subtracted from thedemarcated playing zone 4 are present in the demarcated playing zone 4as free positive unit 6 s. The remaining 7 positive units 6 t to betransported out of the demarcated playing zone 4 are obtained by taking7 null units 10 from within the demarcated playing zone 4, breaking orseparating these 7 null units 10 into their component parts, namely, 7free positive units 6 t and 7 free negative units 8 t, removing the thusobtained 7 free positive units 6 t from the demarcated playing zone 4,and placing the remaining 7 free negative units 8 t in the first 7negative spaces along the axis 12 as shown in FIG. 24.

[0211] With respect to the mathematical operator S₄, the mathematicalexpression X₄, the mathematical operator S₅, and the mathematicalexpression X₅, these mathematical operations are solved in the samemanner as demonstrated in above Example 10 to yield x equals −3.

[0212] It should be noted that, if the above-discussed alternativemethod of treating the signs were employed in solving the variousmathematical operations of above Examples 10 and 11, the answers wouldbe the same.

[0213] As noted with respect to the game apparatus, in the methods ofthe present invention, each null unit 10 preferably comprises onepositive unit 6 reversibly attached to or associated with one negativeunit 8.

[0214] It is also preferred that the methods of the present inventionfurther comprise the step of measuring the number of the free positiveunits 6 and the number of the free negative units 8 within thedemarcated playing zone 4. In one preferred version of the method of thepresent invention, the measurement is performed by the step of placingthe free positive units 6 that are within the demarcated playing zone 4along the positive portion of the axis 12 marked with substantiallyequal spaces from 0 to M and the step of placing the free negative units8 that are within the demarcated playing zone 4 along the negativeportion of the axis 12 marked with substantially equal spaces from 0 toN, where M and N are as defined above. While the positive portion of theaxis 12 preferably forms a continuum with the negative portion of theaxis 12 (as shown in, for example, FIG. 1), the present inventionincludes the embodiment where there is a separate positive axis (notshown) having substantially equal spaces from 0 to M and a separatenegative axis (not shown) having substantially equal spaces from 0 to N,where M and N are as previously defined.

[0215] Another preferred game apparatus 50 of the present invention isshown in FIG. 27. Similar to the game apparatus 1 of FIG. 1, the gameapparatus 50 of FIG. 27 comprises a playing surface 2, a demarcatedplaying environment or zone 4 located within the playing surface 2, aplurality of null units 10 (located within the demarcated playing zone4) removably attached to the playing surface 2, and an axis 52 (locatedwithin the demarcated playing zone 4) numbered from −20 to 20.

[0216] The playing surface 2 of the game apparatus 50 can also be a gameboard, a computer screen, a television screen, a liquid crystal displayscreen, or any other means for displaying the demarcated playing zone 4,the free positive units 6, the free negative units 8, the null units 10,and the axis 52.

[0217] In the version of the invention shown in FIG. 27, the null units10 are also removably attached to the demarcated playing zone 4 bysnuggly fitting into depressions 16 in the demarcated playing zone 4.Likewise, other means for removably attaching the null units 10 to thedemarcated playing zone 4 also include, but are not limited to, Velcro,snap fittings (such as used on clothing or on Lego® building blocks),screw fittings, magnetism, gravity, etc. In addition, when the playingsurface 2 is a computer screen or other essentially two-dimensionalmeans for displaying the playing surface 2, the null units 10 are merelydisplayed on or merely displayed as being associated with theessentially two-dimensional means for displaying the playing surface 2.

[0218] In the embodiment of the invention shown in FIG. 27, the amountof free positive units 6 and free negative units 8 in the demarcatedplaying zone 4 are measured along the axis 52. As shown in FIG. 27, theaxis 52 is numbered from 20 to −20 and is located within the demarcatedplaying zone 4. However, the axis could just as well be numbered from Mto N, where M and N are as defined above. When the playing surface 2 isthe display media used in conjunction with a computerized version of themethod of the present invention, all that need be shown on the displaymedia is the number of free positive units 6 and/or the number of freenegative units 8 within the demarcated playing zone 4.

[0219] As shown in FIG. 27, in addition to the axis 52 located withinthe demarcated playing zone 4, in this embodiment of the invention thereis preferably a second axis 54 located outside the demarcated playingzone 4. The second axis 54 is numbered from 0 to 40, but could just aseasily be numbered from 0 to P, where P is any whole integer. Commonly,P is a positive integer from 10 to 100, more commonly from 20 to 50, andmost commonly from 20 to 40. The purpose of the second axis 54 is to actas a checking point or zone when moving positive units 6 and/or negativeunits 8 into or out of the demarcated playing zone 4. For example, if aproblem called for adding 20 positive units 6 (i.e., +(+20)) to thedemarcated playing zone 4, the 20 positive units 6 could be lined upalong the second axis 54 before being introduced into the demarcatedplaying zone 4 to ensure that the correct number of positive units 6 arebeing transferred into the demarcated playing zone 4. Likewise, if aproblem called for removing 35 negative units 8 (i.e., −(−35)) from thedemarcated playing zone 4, the negative units 8 removed from thedemarcated playing zone 4 could be lined up along the second axis 54 toensure that 35 negative units 8 have, in fact, been removed from thedemarcated playing zone 4.

[0220] As also shown in FIG. 27, in this embodiment of the invention,the game apparatus 50 also comprise a groove 56 for holding the freepositive units 6 and the free negative units 8 that are outside thedemarcated playing zone 4. A divider 58 separates the groove 56 into twoportions, with one portion 60 of the groove 56 being used to hold thefree positive units 6 that are outside of the demarcated playing zone 4and the other portion 62 of the groove 56 being used to hold the freenegative units 8 that are outside the demarcated playing zone 4.

[0221] While the preferred embodiments of the invention have been setforth above in detail, some modifications can be made thereto withoutdeparting from the spirit of the present invention. For example, whilethe above described game apparatuses and methods for adding,subtracting, multiplying, and dividing positive and negative numbers canbe employed to teach students to add and subtract just positive numbers,a simpler device, such as the one shown in FIGS. 10 and 11 can be alsobe used. As shown in FIG. 10, the apparatus 30 comprises a body 32having an opening 34 therein. In the opening 34 is mounted a dowel 36and on the dowel are axially moveable beads 38. The number of axiallymovable beads 38 is generally from 5 to 25 and preferably from 10 to 20.On at least one side of the opening is a scale 40 (such as a ruler orother means for measuring length) for measuring the number of beads 38.The beads 38 can be moved to the left when they are being added and canbe moved to the right when they are being subtracted.

[0222] The reverse side 42 of the apparatus 30 is shown in FIG. 11. Inthis view of this embodiment of the invention, all the elementsmentioned in the preceding paragraph are also present. Accordingly,left- and right-handed students can use the apparatus 30 shown in FIGS.10 and 11 with equal facility.

[0223] Hence, the foregoing alternative embodiments are included withinthe scope of the present invention.

[0224] The apparatuses of the present invention can be made bytechniques well know to those skilled in the art (e.g., injectionmolding, forged, or cast metal, carpentry, etc.). In addition, softwareprograms can be written by those skilled in the art for executing themethods of the present invention and the computerized versions of theinvention can be played on a monitor of any suitably programmableelectrical apparatus (such as a television screen, computer screen,liquid crystal display, etc.).

What is claimed is:
 1. A method for solving a mathematical problem X₁S_(m) X_(m) S_(n) X_(n) . . . S_(z)X_(z) comprising at least onemathematical operation, where the mathematical expressions X₁, X_(m),X_(n), and X_(z) are independently selected from the group consisting ofpositive numbers, negative numbers, multiplication expressions, divisionexpressions, and combinations thereof, the mathematical operators S_(m),S_(n), S_(p), and S_(z) are independently selected from the groupconsisting of the addition operation and the subtraction operation, m isselected from the group consisting of 0 and 2, n is selected from thegroup consisting of 0 and 3, provided that if m is 0, n is 0, and z isselected from the group consisting of 0 and whole integers greater that3, provided that if m is 0, z is 0, the method comprising the step of:(a) starting play in a null state where a demarcated playing zonecomprises a plurality of null units, with each null unit comprising atleast one positive unit reversibly associated with at least one negativeunit, the number of positive and negative units per null unit beingequal.
 2. The method of claim 1 where X₁ is selected from the groupconsisting of positive numbers and negative numbers and the methodfurther comprises a step of selected from the group consisting of:(b)(i) when X₁ is a positive number, step (b)(i) is selected from thegroup consisting of: (A) adding X₁ free positive units to the demarcatedplaying zone; and (B) taking X₁ null units that are within thedemarcated playing zone, separating them into their constituent positiveunits and negative units, and subtracting X₁ negative units from thedemarcated playing zone; and (b)(ii) when X₁ is a negative number, step(b)(ii) is selected from the group consisting of: (A) adding theabsolute value of X₁ free negative units to the demarcated playing zone;and (B) taking the absolute value of X₁ null units that are within thedemarcated playing zone, separating them into their constituent positiveunits and negative units, and subtracting Y₁ positive units from thedemarcated playing zone.
 3. The method of claim 1 where X₁ is amultiplication expression (S_(x)M₁)(S_(y)N₁), where S_(x) and S_(y) areindependently selected from the group consisting a positive sign and anegative sign, M₁ is the absolute value of a number, and N₁ is theabsolute value of a number and the method further comprises the stepselected from the group consisting of: (b)(i) if S_(x) and S_(y) arepositive signs, adding the absolute value of (M₁)(N₁) positive units tothe demarcated playing zone; (b)(ii) if S_(x) is a positive sign andS_(y) is a negative sign, step (b)(ii) is selected from the groupconsisting of: (A) subtracting the absolute value of (M₁)(N₁) positiveunits from the demarcated playing zone by taking the absolute value of(M₁)(N₁) null units that are within the demarcated playing zone,separating them into their constituent positive units and negativeunits, and removing the absolute value of (M₁)(N₁) positive units fromthe demarcated playing zone; and (B) adding the absolute value of(M₁)(N₁) negative units to the demarcated playing zone; (b)(iii) ifS_(x) is a negative sign and S_(y) is a positive sign, step (b)(iii) isselected from the group consisting of: (A) adding the absolute value of(M₁)(N₁) negative units to the demarcated playing zone; and (B)subtracting the absolute value of (M₁)(N₁) positive units from thedemarcated playing zone by taking the absolute value of (M₁)(N₁) nullunits that are within the demarcated playing zone, separating them intotheir constituent positive units and negative units, and removing theabsolute value of (M₁)(N₁) positive units from the demarcated playingzone; and (b)(iv) if S_(x) is a negative sign and S_(y) is a negativesign, subtracting the absolute value of (M₁)(N₁) negative units from thedemarcated playing zone by taking the absolute value of (M₁)(N₁) nullunits that are within the demarcated playing zone, separating them intotheir constituent positive units and negative units, and removing theabsolute value of (M₁)(N₁) negative units from the demarcated playingzone.
 4. The method of claim 1 where X₁ is a division expression(S_(xx)M₁)/(S_(yy)N₁), where S_(xx) and S_(yy) are independentlyselected from the group consisting a positive sign and a negative sign,M₁ is the absolute value of a number, and N₁ is the absolute value of anumber and the method further comprises the step selected from the groupconsisting of: (b)(i) if S_(xx) and S_(yy) are positive signs, addingthe absolute value of (M₁)/(N₁) positive units to the demarcated playingzone; (b)(ii) if S_(xx) is a positive sign and S_(yy) is a negativesign, step (b)(ii) is selected from the group consisting of: (A)subtracting the absolute value of (M₁)/(N₁) positive units from thedemarcated playing zone by taking the absolute value of (M₁)/(N₁) nullunits that are within the demarcated playing zone, separating them intotheir constituent positive units and negative units, and removing theabsolute value of (M₁)/(N₁) positive units from the demarcated playingzone; and (B) adding the absolute value of (M₁)/(N₁) negative units tothe demarcated playing zone; (b)(iii) if S_(xx) is a negative sign andS_(yy) is a positive sign, step (b)(ii) is selected from the groupconsisting of: (A) adding the absolute value of (M₁)/(N₁) negative unitsto the demarcated playing zone; and (B) subtracting the absolute valueof (M₁)/(N₁) positive units from the demarcated playing zone by takingthe absolute value of (M₁)/(N₁) null units that are within thedemarcated playing zone, separating them into their constituent positiveunits and negative units, and removing the absolute value of (M₁)/(N₁)positive units from the demarcated playing zone; and (b)(iv) if S_(xx)is a negative sign and S_(yy) is a negative sign, subtracting theabsolute value of (M₁)/(N₁) negative units from the demarcated playingzone by taking the absolute value of (M₁)/(N₁) null units that arewithin the demarcated playing zone, separating them into theirconstituent positive units and negative units, and removing the absolutevalue of (M₁)/(N₁) negative units from the demarcated playing zone. 5.The method of claim 1 where m is 2, X₂ is selected from the groupconsisting of positive numbers and negative numbers, S₂ is selected fromthe group consisting of an addition operation and a subtractionoperation, and the method further comprises a step selected from thegroup consisting of: (b)(i) when X₂ is a positive number and S₂ is anaddition operation, step (b)(i) comprises adding X₂ free positive unitsto the demarcated playing zone and, if there are any free negative unitswithin the demarcated playing zone, then step (b)(i) further comprisesthe step of combining up to X₂ free negative units that are within thedemarcated playing zone with up to the X₂ free positive units that weremoved into the demarcated playing zone; (b)(ii) when X₂ is a negativenumber and S₂ is an addition operation, step (b)(ii) is selected fromthe group consisting of: (A) adding X₂ free negative units to thedemarcated playing zone and, if there are any free positive units withinthe demarcated playing zone, then step (b)(ii) further comprises thestep of combining up to X₂ free positive units that are within thedemarcated playing zone with up to the X₂ free negative units that weremoved into the demarcated playing zone; and (B) subtracting X₂ freepositive units from the demarcated playing zone and, if there are not X₂free positive units within the demarcated play zone to remove from thedemarcated zone, then step (b)(ii) further comprises the step ofseparating enough null units that are in the demarcated playing zone toobtain up to the required X₂ free positive units and removing the X₂free positive units from the demarcated playing zone; (b)(iii) when X₂is a positive number and S₂ is a subtraction operation, step (b)(iii) isselected from the group consisting of: (A) subtracting X₂ free positiveunits from the demarcated playing zone and, if there are not X₂ freepositive units within the demarcated play zone to remove from thedemarcated zone, then step (b)(iii) further comprises the step ofseparating enough null units that are in the demarcated playing zone toobtain up to the required X₂ free positive units and removing the X₂free positive units from the demarcated playing zone; and (B) adding X₂free negative units to the demarcated playing zone and, if there are anyfree positive units within the demarcated playing zone, then step(b)(iii) further comprises the step of combining up to X₂ free positiveunits that are within the demarcated playing zone with up to the X₂ freenegative units that were moved into the demarcated playing zone; and(b)(iv) when X₂ is a negative number and S₂ is a subtraction operation,step (b)(iv) comprises subtracting X₂ free negative units from thedemarcated playing zone and, if there are not X₂ free negative unitswithin the demarcated play zone to remove from the demarcated zone, thenstep (b)(iv) further comprises the step of separating enough null unitsthat are in the demarcated playing zone to obtain up to the required X₂free negative units and removing the X₂ free negative units from thedemarcated playing zone.
 6. The method of claim 1 where m is 2, X₂ is amultiplication expression (S_(2x)M₂)(S_(2y)N₂), where S_(2x) and S_(2y)are independently selected from the group consisting of a positive signand a negative sign, M₂ is the absolute value of a number, and N₂ is theabsolute value of a number, and S₂ is selected from the group consistingof an addition operation and a subtraction operation, and the methodfurther comprises the steps of: (b)(i) when S_(2x) and S_(2y) arepositive signs and S₂ is an addition operation, step (b)(i) comprisesadding the absolute value of (M₂)(N₂) positive units to the demarcatedplaying zone and, if there are any free negative units within thedemarcated playing zone, then step (b)(i) further comprises the step ofcombining up to the absolute value of (M₂)(N₂) free negative units thatare within the demarcated playing zone with up to the absolute value of(M₂)(N₂) free positive units that were moved into the demarcated playingzone; (b)(ii) when S_(2x) is a positive sign, S_(2y) is a negative sign,and S₂ is an addition operation, step (b)(ii) is selected from the groupconsisting of: (A) subtracting the absolute value of (M₂)(N₂) freepositive units from the demarcated playing zone and, if there are notthe absolute value of (M₂)(N₂) free positive units within the demarcatedplay zone to remove from the demarcated playing zone, then step (b)(ii)further comprises the step of separating enough null units that arewithin the demarcated playing zone into their constituent positive unitsand negative units to obtain up to the required absolute value of(M₂)(N₂) free positive units and removing the absolute value of (M₂)(N₂)free positive units from the demarcated playing zone; and (B) adding theabsolute value of (M₂)(N₂) negative units to the demarcated playing zoneand, if there are any free positive units within the demarcated playingzone, then step (b)(ii) further comprises the step of combining up tothe absolute value of (M₂)(N₂) free positive units that are within thedemarcated playing zone with up to the absolute value of (M₂)(N₂) freenegative units that were moved into the demarcated playing zone;(b)(iii) when S_(2x) is a negative sign, S_(2y) is a positive sign, andS₂ is an addition operation, step (b)(iii) is selected from the groupconsisting of: (A) adding the absolute value of (M₂)(N₂) negative unitsto the demarcated playing zone and, if there are any free positive unitswithin the demarcated playing zone, then step (b)(iii) further comprisesthe step of combining up to the absolute value of (M₂)(N₂) free positiveunits that are within the demarcated playing zone with up to theabsolute value of (M₂)(N₂) free negative units that were moved into thedemarcated playing zone; and (B) subtracting the absolute value of(M₂)(N₂) free positive units from the demarcated playing zone and, ifthere are not the absolute value of (M₂)(N₂) free positive units withinthe demarcated play zone to remove from the demarcated playing zone,then step (b)(iii) further comprises the step of separating enough nullunits that are within the demarcated playing zone into their constituentpositive units and negative units to obtain up to the required absolutevalue of (M₂)(N₂) free positive units and removing the absolute value of(M₂)(N₂) free positive units from the demarcated playing zone; (b)(iv)when S_(2x) and S_(2y) are a negative sign and S₂ is an additionoperation, step (b)(iv) comprises subtracting the absolute value of(M₂)(N₂) free negative units from the demarcated playing zone and, ifthere are not the absolute value of (M₂)(N₂) free negative units withinthe demarcated play zone to remove from the demarcated playing zone,then step (b)(iv) further comprises the step of separating enough nullunits that are within the demarcated playing zone into their constituentpositive units and negative units to obtain up to the required absolutevalue of (M₂)(N₂) free negative units and removing the absolute value of(M₂)(N₂) free negative from the demarcated playing zone; (b)(v) whenS_(2x) and S_(2y) are positive signs and S₂ is a subtraction operation,step (b)(v) comprises subtracting the absolute value of (M₂)(N₂)positive units from the demarcated playing zone and, if there are notthe absolute value of (M₂)(N₂) free positive units within the demarcatedplay zone to remove from the demarcated playing zone, then step (b)(v)further comprises the step of separating enough null units that arewithin the demarcated playing zone into their constituent positive unitsand negative units to obtain up to the required absolute value of(M₂)(N₂) free positive units and removing the (M₂)(N₂) free positiveunits from the demarcated playing zone; (b)(vi) when S_(2x) is apositive sign, S_(2y) is a negative sign, and S₂ is a subtractionoperation, step (b)(vi) is selected from the group consisting of: (A)adding the absolute value of (M₂)(N₂) free positive units to thedemarcated playing zone and, if there are any free negative units withinthe demarcated playing zone, then step (b)(vi) further comprises thestep of combining up to the absolute value of (M₂)(N₂) free negativeunits that are within the demarcated playing zone with up to theabsolute value of (M₂)(N₂) free positive units that were moved into thedemarcated playing zone; and (B) subtracting the absolute value of(M₂)(N₂) free negative units from the demarcated playing zone and, ifthere are not the absolute value of (M₂)(N₂) free negative units withinthe demarcated play zone to remove from the demarcated playing zone,then step (b)(vi) further comprises the step of separating enough nullunits that are within the demarcated playing zone into their constituentpositive units and negative units to obtain up to the required absolutevalue of (M₂)(N₂) free negative units and removing the absolute value of(M₂)(N₂) free negative from the demarcated playing zone; (b)(vii) whenS_(2x) is a negative sign, S_(2y) is a positive sign, and S₂ is asubtraction operation, step (b)(vii) is selected from the groupconsisting of: (A) subtracting the absolute value of (M₂)(N₂) negativeunits from the demarcated playing zone and, if there are not theabsolute value of (M₂)(N₂) free negative units within the demarcatedplay zone to remove from the demarcated playing zone, then step (b)(vii)further comprises the step of separating enough null units that arewithin the demarcated playing zone into their constituent positive unitsand negative units to obtain up to the required absolute value of(M₂)(N₂) free negative units and removing the absolute value of (M₂)(N₂)free negative units from the demarcated playing zone; and (B) adding theabsolute value of (M₂)(N₂) free positive units to the demarcated playingzone and, if there are any free negative units within the demarcatedplaying zone, then step (b)(vii) further comprises the step of combiningup to the absolute value of (M₂)(N₂) free negative units that are withinthe demarcated playing zone with up to the absolute value of (M₂)(N₂)free positive units that were moved into the demarcated playing zone;and (b)(viii) when S_(2x) and S_(2y) are a negative sign and S₂ is asubtraction operation, step (b) comprises adding the absolute value of(M₂)(N₂) free negative units to the demarcated playing zone and, ifthere are any free positive units within the demarcated playing zone,then step (b) further comprises the step of combining up to the absolutevalue of (M₂)(N₂) free positive units that are within the demarcatedplaying zone with up to the absolute value of (M₂)(N₂) free negativeunits that were moved into the demarcated playing zone.
 7. The method ofclaim 1 where m is 2, X₂ is a division expression(S_(2xx)M₂)/(S_(2yy)N₂), where S_(2xx) and S_(2yy) are independentlyselected from the group consisting of a positive sign and a negativesign, M₂ is the absolute value of a number, and N₂ is the absolute valueof a number, and S₂ is selected from the group consisting of an additionoperation and a subtraction operation, and the method further comprisesthe steps of: (b)(i) when S_(2xx) and S_(2yy) are positive signs and S₂is an addition operation, step (b)(i) comprises adding the absolutevalue of (M₂)/(N₂) positive units to the demarcated playing zone and, ifthere are any free negative units within the demarcated playing zone,then step (b)(i) further comprises the step of combining up to theabsolute value of (M₂)/(N₂) free negative units that are within thedemarcated playing zone with up to the absolute value of (M₂)/(N₂) freepositive units that were moved into the demarcated playing zone; (b)(ii)when S_(2xx) is a positive sign, S_(2yy) is a negative sign, and S₂ isan addition operation, step (b)(ii) is selected from the groupconsisting of: (A) subtracting the absolute value of (M₂)/(N₂) freepositive units from the demarcated playing zone and, if there are notthe absolute value of (M₂)/(N₂) free positive units within thedemarcated play zone to remove from the demarcated playing zone, thenstep (b)(ii) further comprises the step of separating enough null unitsthat are within the demarcated playing zone into their constituentpositive units and negative units to obtain up to the required absolutevalue of (M₂)/(N₂) free positive units and removing the absolute valueof (M₂)/(N₂) free positive units from the demarcated playing zone; and(B) adding the absolute value of (M₂)/(N₂) negative units to thedemarcated playing zone and, if there are any free positive units withinthe demarcated playing zone, then step (b)(ii) further comprises thestep of combining up to the absolute value of (M₂)/(N₂) free positiveunits that are within the demarcated playing zone with up to theabsolute value of (M₂)/(N₂) free negative units that were moved into thedemarcated playing zone; (b)(iii) when S_(2xx) is a negative sign,S_(2yy) is a positive sign, and S₂ is an addition operation, step(b)(iii) is selected from the group consisting of: (A) adding theabsolute value of (M₂)/(N₂) negative units to the demarcated playingzone and, if there are any free positive units within the demarcatedplaying zone, then step (b)(iii) further comprises the step of combiningup to the absolute value of (M₂)/(N₂) free positive units that arewithin the demarcated playing zone with up to the absolute value of(M₂)/(N₂) free negative units that were moved into the demarcatedplaying zone; and (B) subtracting the absolute value of (M₂)/(N₂) freepositive units from the demarcated playing zone and, if there are notthe absolute value of (M₂)/(N₂) free positive units within thedemarcated play zone to remove from the demarcated playing zone, thenstep (b)(iii) further comprises the step of separating enough null unitsthat are within the demarcated playing zone into their constituentpositive units and negative units to obtain up to the required absolutevalue of (M₂)/(N₂) free positive units and removing the absolute valueof (M₂)/(N₂) free positive units from the demarcated playing zone;(b)(iv) when S_(2xx) and S_(2yy) are a negative sign and S₂ is anaddition operation, step (b)(iv) comprises subtracting the absolutevalue of (M₂)/(N₂) free negative units from the demarcated playing zoneand, if there are not the absolute value of (M₂)/(N₂) free negativeunits within the demarcated play zone to remove from the demarcatedplaying zone, then step (b)(iv) further comprises the step of separatingenough null units that are within the demarcated playing zone into theirconstituent positive units and negative units to obtain up to therequired absolute value of (M₂)/(N₂) free negative units and removingthe absolute value of (M₂)/(N₂) free negative from the demarcatedplaying zone; (b)(v) when S_(2 xx) and S_(2yy) are positive signs and S₂is a subtraction operation, step (b)(v) comprises subtracting theabsolute value of (M₂)/(N₂) positive units from the demarcated playingzone and, if there are not the absolute value of (M₂)/(N₂) free positiveunits within the demarcated play zone to remove from the demarcatedplaying zone, then step (b)(v) further comprises the step of separatingenough null units that are within the demarcated playing zone into theirconstituent positive units and negative units to obtain up to therequired absolute value of (M₂)/(N₂) free positive units and removingthe (M₂)/(N₂) free positive units from the demarcated playing zone;(b)(vi) when S_(2xx) is a positive sign, S_(2yy) is a negative sign, andS₂ is a subtraction operation, step (b)(vi) is selected from the groupconsisting of: (A) adding the absolute value of (M₂)/(N₂) free positiveunits to the demarcated playing zone and, if there are any free negativeunits within the demarcated playing zone, then step (b)(vi) furthercomprises the step of combining up to the absolute value of (M₂)/(N₂)free negative units that are within the demarcated playing zone with upto the absolute value of (M₂)/(N₂) free positive units that were movedinto the demarcated playing zone; and (B) subtracting the absolute valueof (M₂)/(N₂) free negative units from the demarcated playing zone and,if there are not the absolute value of (M₂)/(N₂) free negative unitswithin the demarcated play zone to remove from the demarcated playingzone, then step (b)(vi) further comprises the step of separating enoughnull units that are within the demarcated playing zone into theirconstituent positive units and negative units to obtain up to therequired absolute value of (M₂)/(N₂) free negative units and removingthe absolute value of (M₂)/(N₂) free negative from the demarcatedplaying zone; (b)(vii) when S_(2xx) (is a negative sign, S_(2yy) is apositive sign, and S₂ is a subtraction operation, step (b)(vii) isselected from the group consisting of: (A) subtracting the absolutevalue of (M₂)/(N₂) negative units from the demarcated playing zone and,if there are not the absolute value of (M₂)/(N₂) free negative unitswithin the demarcated play zone to remove from the demarcated playingzone, then step (b)(vii) further comprises the step of separating enoughnull units that are within the demarcated playing zone into theirconstituent positive units and negative units to obtain up to therequired absolute value of (M₂)/(N₂) free negative units and removingthe absolute value of (M₂)/(N₂) free negative units from the demarcatedplaying zone; and (B) adding the absolute value of (M₂)/(N₂) freepositive units to the demarcated playing zone and, if there are any freenegative units within the demarcated playing zone, then step (b)(vii)further comprises the step of combining up to the absolute value of(M₂)/(N₂) free negative units that are within the demarcated playingzone with up to the absolute value of (M₂)/(N₂) free positive units thatwere moved into the demarcated playing zone; and (b)(viii) when S_(2xx)and S_(2yy) are a negative sign and S₂ is a subtraction operation, step(b) comprises adding the absolute value of (M₂)/(N₂) free negative unitsto the demarcated playing zone and, if there are any free positive unitswithin the demarcated playing zone, then step (b) further comprises thestep of combining up to the absolute value of (M₂)/(N₂) free positiveunits that are within the demarcated playing zone with up to theabsolute value of (M₂)/(N₂) free negative units that were moved into thedemarcated playing zone.
 8. The method of claim 1 further comprising atleast one of the following steps of: (b) measuring the number of thefree positive units within the demarcated playing zone; and (c)measuring the number of the free negative units within the demarcatedplaying zone.
 9. The method of claim 1 further comprising at least oneof the following steps: (b) placing the free positive units that arewithin the demarcated playing zone along the positive portion of an axismarked with substantially equal spaces from 0 to M; and (c) placing thefree negative units that are within the demarcated playing zone alongthe negative portion of the axis marked with substantially equal spacesfrom 0 to N, with M being a positive whole integer and N being anegative whole integer.
 10. The method of claim 1 where each null unitcomprises one positive unit reversibly associated with one negativeunit.
 11. The method of claim 1 where each null unit comprises onepositive unit reversibly associated with one negative unit and themethod further comprising at least one of the following steps: (b)placing the free positive units that are within the demarcated playingzone along the positive portion of an axis marked with substantiallyequal spaces from 0 to M; and (c) placing the free negative units thatare within the demarcated playing zone along the negative portion of theaxis marked with substantially equal spaces from 0 to N, with M being apositive whole integer and N being a negative whole integer.
 12. Themethod of claim 1 where m is 2 and n is
 3. 13. The method of claim 1where m is 2, n is 3, and z is a whole integer greater than
 3. 14. Themethod of claim 1 further comprising: (b) at least one step selectedfrom the group consisting of: (A) solving each multiplication expressionin the mathematical problem; and (B) solving each division expression inthe mathematical problem; (c) at least one step selected from the groupconsisting of: (A) substituting the solution to each multiplicationexpression into the mathematical problem to form a revised mathematicalproblem; and (B) substituting the solution to each division expressioninto the mathematical problem to form a revised mathematical problem;and (d) the step of solving the revised mathematical problem.
 15. Themethod of claim 1 further comprising the step of sequentially solvingeach mathematical operation in the mathematical problem.
 16. The methodof claim 1 further comprising at least one of the following additionalthe steps of: (b) measuring the number of the free positive units to beadded to the demarcated playing zone; (c) measuring the number of thefree negative units to be added to the demarcated playing zone; (d)measuring the number of the free positive units that have been removedfrom the demarcated playing zone; and (e) measuring the number of thefree negative units that have been removed from the demarcated playingzone.
 17. A method for solving a mathematical problem comprising atleast one mathematical operation selected from the group consisting ofaddition, subtraction, multiplication, division, and combinationsthereof, the method comprising at least one step selected from the groupconsisting of: (a) adding at least one positive unit to a demarcatedplaying zone; (b) subtracting at least one positive unit from thedemarcated playing zone; (c) adding at least one negative unit to thedemarcated playing zone; (d) subtracting at least one negative unit fromthe demarcated playing zone; (e) separating at least one null unit intothe null unit's constituent parts, where each null unit (i) comprises anequal number of positive units and negative units and (ii) is located inthe demarcated playing zone; (f) combining at least one free negativeunit with at least one free positive unit in the demarcated playing zoneto form at least one null unit; and (g) combinations of steps (a)through (f).
 18. The method of claim 17 comprising at least two stepsselected from the group consisting of steps (a) through (f).
 19. Themethod claim 18 further comprising the step of: (h) starting play in anull state where the demarcated playing zone comprises a plurality ofnull units, with each null unit comprising at least one positive unitreversibly associated with at least one negative unit, the number ofpositive and negative units per null unit being equal.
 20. The methodclaim 17 further comprising the step of: (h) starting play in a nullstate where the demarcated playing zone comprises a plurality of nullunits, with each null unit comprising at least one positive unitreversibly associated with at least one negative unit, the number ofpositive and negative units per null unit being equal.